Doppler radars compare echoes from two pulses sent shortly after another, so that the comparison is usually based on a difference in phase. If the object reflecting the pulse moved by a distance smaller than a quarter the wavelength of the radar beam, the measurement reveals its radial speed relative to the radar antenna. For instance, if the radar transmission has a frequency of 5 GHz (wavelength 6 cm) and transmitted pulses follow each other at one millisecond intervals, the maximum distance traveled is 1.5 cm and the absolute value of the maximum detectable speed is 15 m/s. If the scattering object moved faster, the speed measurement will be wrong by a multiple of 30 m/s.
A weather radar differs from e.g. a traffic radar in that it will not measure the single speed of a single object but the distribution of speeds of scattering targets, also called just scatterers for short. In meteorological terms, the scatterers are radiation-scattering hydrometeors (solid particles and/or liquid droplets) within a volume of air. Depending on the resolution of the radar this so-called target volume can be very large, for instance one cubic kilometer. All scatterers within the target volume will not move at a uniform speed; for example a cloud will constantly change its shape. If there are several layers of clouds, they often move with different speeds into different directions. The weather radar typically measures several differently moving layers in the atmosphere simultaneously. The size of the target volume is defined by the dimensions (width) of the radar beam as well as the resolution in units of time that is used to process the radar measurements. Even if the radiation-scattering targets are actually solid particles and/or liquid droplets and not the gaseous constituents of air, it is conventional to say for example that the weather radar may measure “the movements of air” or “wind speed”.
In the simplest case a radar emits pulses at a fixed frequency. If the radar is sending a single pulse every millisecond, its Pulse Repetition Frequency (PRF) is 1 kHz. A more sophisticated transmission scheme is Multi-PRI (Multiple Pulse Repetition Intervals), which uses a periodic pattern of pulses. An example is a period of 6 ms that involves sending pulses at 0 ms, 1 ms, and 3 ms. The pauses between pulses would be 1 ms, 2 ms and 3 ms, then starting over. FIG. 1 illustrates such a sample pattern of pulses. Simultaneous Multiple Pulse Repetition Frequency (SMPRF) is Multi-PRI which does not wait for the pulse to fade before sending the next pulse. As a result the SMPRF radar will receive echoes of multiple pulses at the same time.
An autocorrelation function, ACF for short, is a conventional way to explain and measure the remotely sensed movements of air. The ACF shows how the echo of a volume of air changes over time. It can be considered as an indicator of how a time-shifted copy of the received signal correlates with the original received signal. Values of the ACF are complex numbers. An ACF value equal to the real number “1” (with the imaginary part equal to zero) means simply that the echo did not change at all. Here we assume that the ACF values are scaled correspondingly: the general level of unscaled ACF values depends on the overall reflectivity of the target, i.e. on how much of the radar signal was reflected back.
Values of the ACF may be plotted as a function of the time shift between the compared signals. At time shift zero the properly scaled ACF will be always 1 (because any arbitrary signal correlates perfectly with its exact copy). If the echo changes, the absolute value of the ACF will decrease. An absolute value 0 of the ACF means no correlation or similarity at all. If the volume of air seems to be moving but stays otherwise the same, the ACF will “turn” in the complex plane.
The ACF does not say anything about the shape of the reflecting object or how the echo actually looks like. It explains the change. For instance, the radar echoes of a perfectly flat wall and a mountain will be entirely different. But since they both stand still they have the same ACF (of a constant 1).
If the speed distribution in the volume consists of more than one peak value, each with some spread around the peak, the ACF doesn't have to descend continuously. FIGS. 2 and 3 demonstrate an idealised weather condition with two layers of clouds moving at different speeds. In FIG. 2 the solid line illustrates the absolute value, the dotted line the real part, and the dashed line the imaginary part of the ACF. FIG. 3 shows the corresponding histogram of speeds with maxima at 6 m/s and 10 m/s. The choice of scale for the vertical axis will be explained in more detail later. For the moment it suffices to note that it has a certain association with the scaling of the ACF; the total “mass” (i.e. the sum of all non-zero values) in the histogram reflects the true, unscaled value of ACF's zero lag (supposing that the histogram doesn't have any error).
An ACF illustrated with a smooth, continuous line like in FIG. 2 is a mathematical construction. A practical radar does not measure the entire function. A conventional Doppler radar, for example, may give just one meaningful complex number as an output. FIG. 4 shows the ACF of a model cloud over 0.02 seconds. The solid line is the absolute value of the ACF, the dotted line is the real part and the dashed line is the imaginary part. A conventional Doppler radar will measure only two data points, one at time shift zero and the other at 5 ms. FIG. 5 shows the real part of the measurement as a plus sign and the imaginary part as an x. The measurement at time shift zero is always equal to the real number 1, so it is not explicitly shown in the illustration. The output of the conventional Doppler radar does not enable the calculation of any speed distributions; it only gives the single most typical value. If an aeroplane or some other solid, efficient scatterer happened to appear in the radar beam, the whole attempt to measure wind speed is ruined.
Using Multi-PRI or SMPRF gives more information about the ACF: more data points are measured, and a reasonable ACF can be illustrated. As an example we may consider a pattern of 3 pulses, with the intervals between pulses being 1750 μs, 2000 μs, and 2500 μs respectively. The total period is 6250 μs. When comparing the echo of the first and second pulses, one gets a measurement for the change in the cloud during 1750 μs. When comparing the echo of the fourth and fifth pulses we get another one. And another one for the seventh and eighth, and so on. Adding them up (integrating) gives a statistical result for a “lag” of 1750 μs. Measurements for 2000 μs and 2500 μs combine in the same way respectively. Comparing the first echo with the third will result in a measurement for a lag of 3750 μs (1750+2000). Comparing the third and the fifth will result in a measurement for 4250 μs. Second and fourth gives 4500 μs. First and fourth give 6250 μs, and so on. This can be continued as long as desired, but very long lags make sense only if there is little turbulence.
As a result, a number of data points are measured for the ACF. FIG. 6 shows 21 nontrivial data points (the trivial one being the one at time shift zero). The real part of the data points are shown as plus signs and the imaginary parts are shown as x signs.
Concerning noise, the ACF value at lag zero is again one after normalization. When calculating other lags for pure noise, the random phase of Multi-PRI and SMPRF pulses will result in a complex number of random phase and an absolute value of about one. The integration of the pulses will now add up complex numbers of random phase, resulting in a random walk. Assuming gaussian noise we can expect the absolute value of any lag to be the inverse of the square root of the number of elements added in the integration. For integration of 100 pulses, the ACF value at any lag is thus expected to be a complex number of absolute value of 0.1. Assuming the pulse sequence described above, the lags at 6250, 12500 and 18750 μs integrate three times the number of measurements, resulting in an absolute value of 0.06. FIG. 7 shows 21 data points of the expected absolute values. The absolute value at lag 0 is 1, but it is not explicitly shown in the illustration. The real and imaginary parts do not have an expected value and are not shown in the illustration.
Measured ACFs that look similar to the graph of FIG. 7 should be discarded and the measurement should be marked as pure noise. If individual points of a measured ACF have an absolute value comparable to noise, they can be ignored individually.
The drawbacks of prior art have been related to the relatively uncommon use of Multi-PRI and SMPRF elsewhere than in ionospheric research, as well as to the large requirements of computing capacity that has been needed to turn measured ACFs into meaningful indications of speed distribution.